Correction Capabilities of the Reed-Solomon Codes Decoded with the Guruswami-Sudan Algorithm (original) (raw)
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Two new decoding algorithms for Reed-Solomon codes
Applicable Algebra in Engineering, Communication and Computing, 1994
The subject of decoding Reed-Solomon codes is considered, By reformulating the Berlekamp and Welch key equation and introducing new versions of this key equation, two new decoding algorithms for Reed-Solomon codes will be presented. The two new decoding algorithms are significant for three reasons. Firstly the new equations and algorithms represent a novel approach to the extensively researched problem of decoding Reed-Solomon codes. Secondly the algorithms have algorithmic and implementation complexity comparable to existing decoding algorithms, and as such present a viable solution for decoding Reed-Solomon codes. Thirdly the new ideas presented suggest a direction for future research. The first algorithm uses the extended Euclidean algorithm an~t is very efficient for a systolic VLSI implementation. The second decoding algorithm presented is similar in nature to the original decoding algorithm of Peterson except that the syndromes do not need to be computed and the remainders are used directly. It has a regular structure and will be efficient for implementation only for correcting a small number of errors. A systolic design for computing the Lagrange interpolation of a polynomial, which is needed for the first decoding algorithm, is also presented.
New Approaches to the Analysis and Design of Reed-Solomon Related Codes
2007
The research that led to this thesis was inspired by Sudan's breakthrough that demonstrated that Reed-Solomon codes can correct more errors than previously thought. This breakthrough can render the current state-of-the-art Reed-Solomon decoders obsolete. Much of the importance of Reed-Solomon codes stems from their ubiquity and utility. This thesis takes a few steps toward a deeper understanding of Reed-Solomon codes as well as toward the design of efficient algorithms for decoding them. After studying the binary images of Reed-Solomon codes, we proceeded to analyze their performance under optimum decoding. Moreover, we investigated the performance of Reed-Solomon codes in network scenarios when the code is shared by many users or applications. We proved that Reed-Solomon codes have many more desirable properties. Algebraic soft decoding of Reed-Solomon codes is a class of algorithms that was stirred by Sudan's breakthrough. We developed a mathematical model for algebraic so...
Error Correcting Codes: Combinatorics, Algorithms and Applications 1 Reed-solomon Codes
2007
We begin with the definition of Reed-Solomon codes. Definition 1.1 (Reed-Solomon code) . Let Fq be a finite field andFq[x] denote theFq-space of univariate polynomials where all the coefficients of x are fromFq. Pick {α1, α2, ...αn} distinct elements (also calledevaluation points ) of Fq and choosen and k such thatk ≤ n ≤ q. We define an encoding function for Reed-Solomon code as RS : Fq → F n q as follows. A message m = (m0, m1, ..., mk−1) with mi ∈ Fq is mapped to a degree k − 1 polynomial.
A new decoding algorithm for correcting both erasures and errors of reed-solomon codes
IEEE Transactions on Communications, 2003
In this paper, a high efficient decoding algorithm is developed here in order to correct both erasures and errors for Reed-Solomon (RS) codes based on the Euclidean algorithm together with the Berlekamp-Massey (BM) algorithm. The new decoding algorithm computes the errata locator polynomial and the errata evaluator polynomial simultaneously without performing polynomial divisions, and there is no need for the computation of the discrepancies and the field element inversions. Also, the separate computation of the Forney syndrome needed in the decoder is completely avoided. As a consequence, the complexity of this new decoding algorithm is dramatically reduced. Finally, the new algorithm has been verified through a software simulation using C ++ language. An illustrative example of (255,239) RS code using this program shows that the speed of the decoding process is approximately three times faster than that of the inverse-free Berlekamp-Massey algorithm.
Decoding of Reed Solomon Codes beyond the Error-Correction Bound
Journal of Complexity, 1997
We present a randomized algorithm which takes as input n distinct points f(x ; y )g from F 2 F (where F is a field) and integer parameters t and d and returns a list of all univariate polynomials f over F in the variable x of degree at most d which agree with the given set of points in at least t places (i.e., y = f (x ) for at least t values of i), provided t = ( p nd). The running time is bounded by a polynomial in n. This immediately provides a maximum likelihood decoding algorithm for Reed Solomon Codes, which works in a setting with a larger number of errors than any previously known algorithm. To the best of our knowledge, this is the first efficient (i.e., polynomial time bounded) algorithm which provides error recovery capability beyond the error-correction bound of a code for any efficient (i.e., constant or even polynomial rate) code.
A Fast Algorithm for the Syndrome Calculation in Algebraic Decoding of Reed–Solomon Codes
IEEE Transactions on Communications, 2007
In this letter, Fedorenko and Trifonov's procedure is applied to evaluate the syndrome of the received word in timedomain Reed-Solomon decoders. This application leads to a substantial reduction of the computational complexity of the syndrome polynomial for correcting both errors and erasures. Moreover, simulation results for this new syndrome method are given.
Efficient Frequency-Domain Decoding Algorithms for Reed-Solomon Codes
2015
Recently, a new polynomial basis over binary extension fields was proposed such that the fast Fourier transform (FFT) over such fields can be computed in the complexity of order O(n lg(n)), where n is the number of points evaluated in FFT. In this work, we reformulate this FFT algorithm such that it can be easier understood and be extended to develop frequencydomain decoding algorithms for (n = 2 m , k) systematic Reed-Solomon (RS) codes over F2m , m ∈ Z + , with n − k a power of two. First, the basis of syndrome polynomials is reformulated in the decoding procedure so that the new transforms can be applied to the decoding procedure. A fast extended Euclidean algorithm is developed to determine the error locator polynomial. The computational complexity of the proposed decoding algorithm is O(n lg(n − k) + (n − k) lg 2 (n − k)), improving upon the best currently available decoding complexity O(n lg 2 (n) lg lg(n)), and reaching the best known complexity bound that was established by Justesen in 1976. However, Justesen's approach is only for the codes over some specific fields, which can apply Cooley-Tucky FFTs. As revealed by the computer simulations, the proposed decoding algorithm is 50 times faster than the conventional one for the (2 16 , 2 15) RS code over F 2 16 .
Fast parallel algorithms for decoding Reed-Solomon codes based on remainder polynomials
Information Theory, IEEE Transactions on, 1995
The problem of decoding cyclic error correcting codes is one of solving a constrained polynomial congruence, often achieved using the BerlekampMassey or the extended Euclidean algorithm on a key equation involving the syndrome polynomial. A module-theoretic approach to the solution of polynomial congruences is developed here using the notion of exact sequences. This technique is applied to the Welch-Berlekamp key equation for decoding ReedSolomon codes for which the computation of syndromes is not required. It leads directly to new and efficient parallel decoding algorithms that can be realized with a systolic array. The architectural issues for one of these parallel decoding algorithms are examined in some detail. Index Tenns-ReedSolomon codes, decoding algorithms, systolic arrays, Welch-Berlekamp equations, modules.
A Simplified Step-by-Step Decoding Algorithm for Parallel Decoding of Reed–Solomon Codes
IEEE Transactions on Communications, 2000
A simplified parallel step-by-step decoding algorithm is proposed for decoding Reed-Solomon (RS) codes. It uses new method to calculate the determinants of the temporarily changed syndrome matrices, based on the property of these matrices determined in this paper. By using the proposed method, the calculations of the determinants of the temporarily changed syndrome matrices become much simpler and thus the computational complexity of the step-by-step decoding algorithm is significantly reduced.
INTERNATIONAL JOURNAL OF ENGINEERING DEVELOPMENT AND RESEARCH (IJEDR) (ISSN:2321-9939), 2014
Reed-Solomon codes are very useful and effective in burst error in noisy environment. In decoding process for 1 error or 2 errors create easily with using procedure of Peterson-Gorenstein –Zierler algorithm. If decoding process for 3 or more errors, these errors can be solved with key equation of a new algorithm named Berlekamp-massey algorithm. In this paper, wide discussion of procedures of Peterson-Gorenstein –Zierler algorithm and Berlekamp-Massey algorithm and show the advantages of modified version of Berlekam-Massey algorithm with its steps.